On six-dimensional AH-submanifolds of class W1⊕W2⊕W4 in Cayley algebra

Six-dimensional submanifolds of Cayley algebra equipped with an almost Hermitian structure of class W1 W2 W4 defined by means of three-fold vector cross products are considered. As it is known, the class W1 W2 W4 contains all Kählerian, nearly Kählerian, almost Kählerian, locally conformal Kählerian, quasi-Kählerian and Vaisman — Gray manifolds. The Cartan structural equations of the W1 W2 W4 -structure on such six-dimensional submanifolds of the octave algebra are obtained. A criterion in terms of the configuration tensor for an arbitrary almost Hermitian structure on a six-dimensional submanifold of Cayley algebra to belong to the W1 W2 W4 -class is established. It is proved that if a six-dimensional W1 W2 W4 -submanifold of Cayley algebra satisfies the quasi-Sasakian hypersurfaces axiom (i.e. a hypersurface with a quasi-Sasakian structure passes through every point of such submanifold), then it is an almost Kählerian manifold. It is also proved that a six-dimensional W1 W2 W4 -submanifold of Cayley algebra satisfies the eta-quasi-umbilical quasi-Sasakian hypersurfaces axiom, then it is a Kählerian manifold.

On nonexistence of Kenmotsu structure on аст-hypersurfaces of cosymplectic type of a Kählerian manifold

Almost contact metric (аст-)structures induced on oriented hypersurfaces of a Kählerian manifold are considered in the case when these аст- structures are of cosymplectic type, i. e. the contact form of these structures is closed. As it is known, the Kenmotsu structure is the most important non-trivial example of an almost contact metric structure of cosymplectic type. The Cartan structural equations of the almost contact metric structure of cosymplectic type on a hypersurface of a Kählerian manifold are obtained. It is proved that an almost contact metric structure of cosymplectic type on a hypersurface of a Kählerian manifold of dimension at least six cannot be a Kenmotsu structure. Moreover, it follows that oriented hypersurfaces of a Kählerian manifold of dimension at least six do not admit non-trivial almost contact metric structures of cosymplectic type that belong to any well studied class of аст-structures. The present results generalize some results on almost contact metric structures on hypersurfaces of an almost Hermitian manifold obtained earlier by V. F. Kirichenko, L. V. Stepanova, A. Abu-Saleem, M. B. Banaru and others.

SPECIAL HERMITIAN MANIFOLDS AND THE 1-COSYMPLECTIC HYPERSURFACES AXIOM

It is proved that if a special Hermitian manifold complies with the 1-cosymplectic hypersurfaces axiom, then it is a Kählerian manifold.

Sasakian Hypersurfaces of the Generalized Concircular Recurrent Kahlerian Manifold

In this paper we consider a recurrent sasakian hyper surface of the generalized con circular recurrent Kahlerian manifold and determine some conditions on the vector fields used in the sasakian structure. Further we determine such conditions for ϕ sasakian hyper surface also.

FAMILIES OF HOLOMORPHIC BUNDLES

The first goal of the article is to solve several fundamental problems in the theory of holomorphic bundles over non-algebraic manifolds. For instance, we prove that stability and semi-stability are Zariski open properties in families when the Gauduchon degree map is a topological invariant, or when the parameter manifold is compact. Second, we show that, for a generically stable family of bundles over a Kähler manifold, the Petersson–Weil form extends as a closed positive current on the whole parameter space of the family. This extension theorem uses classical tools from Yang–Mills theory (e.g., the Donaldson functional on the space of Hermitian metrics and its properties). We apply these results to study families of bundles over a Kählerian manifold Y parametrized by a non-Kählerian surface X, proving that such families must satisfy very restrictive conditions. These results play an important role in our program to prove existence of curves on class VII surfaces [22–24].

Harmonic Maps and Stability onf-Kenmotsu Manifolds

The purpose of this paper is to study some submanifolds and Riemannian submersions on anf-Kenmotsu manifold. The stability of aϕ-holomorphic map from a compactf-Kenmotsu manifold to a Kählerian manifold is proven.

Skew-symmetric vector fields on aCR-submanifold of a para-Kählerian manifold

We deal with aCR-submanifoldMof a para-Kählerian manifoldM˜, which carries aJ-skew-symmetric vector fieldX. It is shown thatXdefines a global Hamiltonian of the symplectic formΩonM⊤andJXis a relative infinitesimal automorphism ofΩ. Other geometric properties are given.

Three theorems on cosymplectic hypersurfaces of six-dimensional Hermitian submanifolds of the Cayley algebra

It is proved that cosymplectic hypersurfaces of six-dimensional Hermitian submanifolds of the octave algebra are ruled manifolds. A necessary and sufficient condition for a cosymplectic hypersurface of a Hermitian submanifoldM6⊂Oto be a minimal submanifold ofM6is established. It is also proved that a six-dimensional Hermitian submanifoldM6⊂Osatisfying theg-cosymplectic hypersurfaces axiom is a Kählerian manifold.

ON THE CALABI ENERGY OF EXTREMAL KÄHLER METRICS

Fix a positive (1, 1)-class on a compact Kählerian manifold. Given a Kähler form representing this class, define its Calabi energy to be the L2-norm of its scalar curvature. This note proves that a critical metric for the Calabi energy, if any, is a global minimum among representatives of the chosen class, and that the critical value is determined a priori by the Kähler class. This answers affirmatively two questions of Calabi ([2, p. 99]).